What is 0.4 repeating as a fraction?

Understand the Problem

The question is asking how to express the decimal 0.4, with the digit '4' repeating indefinitely, as a fraction. To solve this, we can use algebraic methods to convert the repeating decimal into a fraction.

Answer

The fraction that represents $0.4444\ldots$ is $\frac{4}{9}$.

Steps to Solve

Let the repeating decimal be a variable

Let $x$ be the repeating decimal, so we set: $$ x = 0.4444\ldots $$

Multiply by a power of 10

To eliminate the repeating part, multiply both sides of the equation by 10: $$ 10x = 4.4444\ldots $$

Set up a subtraction equation

Now, we have two equations:

( x = 0.4444\ldots )
( 10x = 4.4444\ldots )

Subtract the first equation from the second: $$ 10x - x = 4.4444\ldots - 0.4444\ldots $$

Simplify the equation

This simplifies to: $$ 9x = 4 $$

Solve for ( x )

Now, divide both sides by 9 to solve for ( x ): $$ x = \frac{4}{9} $$

The repeating decimal (0.4444\ldots) can be expressed as the fraction (\frac{4}{9}).

More Information

The fraction (\frac{4}{9}) represents the repeating decimal (0.4444\ldots). While this fraction cannot be simplified further, it accurately depicts the value of the original decimal. Repeating decimals can always be converted into fractions using algebraic methods like these.

Tips

Forgetting to subtract the original equation from the multiplied equation, which is essential to isolate the variable. Always ensure you set up the subtraction correctly.

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